difference between idempotent and projection operators in book of conway, functional analysis, section operators on Hilbert space(projection and idempotent) say that a projection is an idempotent such P that $(kerP)=(rangP)^\perp$.
but from the next proposition part c, we can conclude that for every idempotent we have $(kerP)=(rangP)^\perp$. am i right? 
if this is true then what is the difference between projection and idempotent? please help me. thanks.
 A: The notion of idempotent does not require a Hilbert space structure, you simply require an operator, say acting on a Banach space, to satisfy $T^2 = T$. So $T$ restricted to $Ran(T)$ is the identity.
On the other hand, when you write $^{\perp}$, implicitly there is a Hilbert space structure. Then your condition $Ker (P) = Ran (P)^{\perp}$ means $P$ is a self-adjoint idempotent, i.e. (orthogonal) projection on Hilbert space. 
A: Proposition 3.2c in Conway's book indeed says that for every idempotent $E$ acting on a Hilbert space $\mathscr{H}$ there holds
$$\mathscr{H} = \mathscr{M}+\mathscr{N} \tag{1}$$
where $\mathscr{M}=\operatorname{ran}E$ and $\mathscr{N}=\operatorname{ker}E $. 
If in  (1), "$+$" meant orthogonal sum, your reasoning would be correct. But this is not what Conway means. He uses $\oplus$ for orthogonal sum, see section 1.6, or later in the section on projections. The meaning of $+$ here is the vector sum of sets: 
$$A+B = \{a+b:a\in A,b\in B\}$$
So, (1) simply says that every element of $\mathscr{H}$ can be written as the sum of two vectors, one in $\mathscr{M}$ and the other in $\mathscr{N}$. This is quite a weak statement; it does not even exclude the possibility that two spaces could overlap, or be the same: 
$$\mathscr{H} = \mathscr{H}+\mathscr{H}  $$ is a true statement. This is why Conway also puts $\mathscr{M} \cap  \mathscr{N} = (0)$ as a part of 3.2c.
Just for the sake of completeness, a simple example:   $$\begin{pmatrix} 1 & 0 \\ 1 & 0\end{pmatrix}$$
in two-dimensional space, like $\mathbb R^2$. This is an idempotent operator; the kernel is the vertical axis, the range is the diagonal.
